It is always best to isolate the barrel so the check ring is the only thing being tested, and outside influences are minimized.
With a 2-shot machine, how do I perform a check ring capability study on my 2nd shot?
You want to run the shot by itself if possible. Using both barrels will result in a less reliable number. If you were forced to use both shots, the fill and pack the first shot and then subtract that weight from the final weight.
When using a two shot machine, have a way to segregate the parts from the two cores. If the check ring shows high variation, you can run the numbers again separately.
For example if the combined variation is 5%, but the variation in shot 1 and shot 2 may only be 1% by themselves. In such a case, the ring is fine, but the tooling needs help.
Adversely if a high variation is present in one or both of the shots when calculated alone, then the check ring is inconsistent.